Theory, Methods, and Applications of Fractional CalculusView this Special Issue
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Stability, Boundedness, and Lagrange Stability of Fractional Differential Equations with Initial Time Difference
Differential inequalities, comparison results, and sufficient conditions on initial time difference stability, boundedness, and Lagrange stability for fractional differential systems have been evaluated.
The problem of stability of solutions is one of the major problems in the theory of differential equations. Lyapunov function and the Lyapunov direct method allow us to obtain sufficient conditions for the stability of a system without explicitly solving the differential equations [1, 2]. The method generalizes the idea which shows that the system is stable if there are some Lyapunov function candidates for the system.
Only a few decades ago, it was realized that fractional calculus provides an attractive tool for modelling the real world problems. The differentiation and integration of arbitrary orders have found applications in diverse fields of science and engineering like viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals [3–5]. Recently, some attention has been drawn on stability analysis of fractional differential equations (FDE) [6–9].
In practical situations, it is possible to have not only a change in initial position but also in initial time because of all kinds of disturbed factors. When we do consider such a deviation in initial time, it causes measuring the difference between any two different solutions starting with different initial times. From this point of view, several studies have been made on this problem to explore the stability and boundedness, criteria for differential systems relative to initial time difference (ITD) by using variation of parameters and differential inequalities technique [10–13]. In this paper, the stability and boundedness criteria for FDE relative to initial time difference have been investigated by using comparison method. In Section 2 the differences between classical notion of stability and the notion of initial time difference (ITD) stability have been discussed and compared by giving basic definitions. In Section 3 new differential inequalities and comparison results relative to initial time difference are obtained. Then, we investigate ITD stability, boundedness and Lagrange stability by using results obtained in Section 3. Lastly the conclusions are given in Section 4.
2. Fractional Calculus
Fractional calculus generalizes the derivative and the integral of a function to a noninteger order [3, 4, 14]. Although there are several definitions of fractional derivatives and fractional integrals, only the relative ones are given below. Let be a function.
Definition 1. The fractional integral (or the Riemann-Liouville (RL) type integral) of order is defined as
Definition 2. The RL fractional derivatives of order of are defined as
Definition 3. The Caputo fractional derivative of order of is defined as To simplify the notations we will use , , and instead of , , and , respectively. There are some relation between these fractional integral and derivatives. For details please see [3, 4, 14].
Property 1. (i).(ii), where represents .(iii), where represents .(iv) and .(v) and .(vi).(vii) and , where is arbitrary constant.
After giving definition and properties of fractional integral and derivatives, we consider fractional order IVP with (RL) and Caputo derivative, respectively; where , and . Then IVP's are equivalent to the following Volterra fractional integral equations: respectively. In  authors develop a relation between the solutions of Caputo fractional differential equations and those of (RL) fractional differential equations.
3. Stability versus Initial Time Difference Stability
Consider the IVP for the system of nonlinear fractional differential equation where , and . Suppose that the function is smooth enough to guarantee existence, uniqueness, and continuous dependence of solutions of IVP (7). Denote by the solution of (7). Consider also the initial value problem at a different initial data; that is, let and be the solution of the system (7). Assume that is the solution on which we shall study stability and boundedness criteria with respect to it. Set and denote .
Before giving our comparison theorem, stability, boundedness criteria, and Lagrange stability for FDE we need to introduce the following definitions.
Definition 4. The solution of (7) is said to be(S1)stable with ITD, if given and ; there exist and such that (S2)uniformly stable with ITD, if holds with and independent of .
Definition 5. The system (7) is said to be(B1)equibounded with ITD, if given and ; there exist and such that (B2)uniformly bounded with ITD, if holds with and independent of ;(A1)attractive in the large with ITD, if for every and each there exists and such that (L1)Lagrange stable if (B1) and (A1) hold.
Definition 6. A function is said to belong to the class such that
Definition 7. We define the generalized derivative with respect to the system (7) as follows:
for , where is the solution of (7) and , where is the solution of (7) and .
The stability with ITD gives us an opportunity to compare solutions of FDE where both initial time and position are different. In the case of differential equation, stability with ITD is studied in [10–13, 16]. We will give a brief overview of both concepts of stability.
Case 1 (classical notion of stability). Let be a solution of (7). Study the stability of . Consider the solution of (7). Set the IVP as
where . The function is a solution of the IVP (13).
The IVP (13) has a zero solution and the study of stability properties of the nonzero solution of (7) is reduced to the stability of the zero solution of transformed system (13).
Case 2 (stability with ITD). Study the stability with initial time difference of . Consider the solution of (7) with different initial data as . Set the IVP as where . Then is a solution of (14). The system (14) has no zero solution since . Therefore, in this case the study of stability with ITD of could not be reduced to the study of stability of the zero solution of an appropriate fractional differential system.
4. Main Results
4.1. Differential Inequalities and Comparison Results
4.1.1. Differential Inequalities
Definition 8 (see ). is said to be continuous; that is, , if and only if the Caputo derivative of exists and satisfies
Lemma 9. Let . Suppose that for any , one has and for ; then it follows that
Proof. From the relation between Riemann-Liouville and Caputo fractional derivatives we write Since we have . Therefore, we obtain Finally using the lemma for R-L derivative from  and (18) implies that
Theorem 10. Let , and Then implies
Proof. Suppose that relation (21) is false. Then since and , are continuous, there exists a with and for . Set . Then and for . Hence, the hypothesis of Lemma 9 holds and we conclude that , which means that . Consider which is a contradiction. Thus the conclusion of the theorem holds and the proof is complete.
Theorem 11. Let , and Assume satisfies the Lipschitz condition Then, implies
Proof. We set , where is the solution of linear fractional differential equation , . Then from , we get . In order to get (25), we need to apply Theorem 10 to and . Using and Lipschitz condition we have
Applying now Theorem 10 to and , we get for every and consequently making , we get the desired estimate (25).
4.1.2. Comparison Results
The most commonly used technique in the theory of differential equations is related to the estimation of a function satisfying a differential inequality by the extremal solutions of the related differential equation. The following theorems give such estimate with initial time difference.
Theorem 12. Assume that and where . Let be the maximal solution of the IVP existing on such that . Then one has
Proof. In view of the definition of the maximal solution , it is enough to prove that , , where is any solution of the IVP Using Lemma 9, we get . And from uniformly on each compact set , we get the desired estimate (29).
Theorem 13. Assume that(i), and (ii)the maximal solution of the IVP exists for ;(iii) is nondecreasing in for each and .Then (a) , and (b) , .
Proof. (a) It is well known that if is any solution of
sufficiently small then on every compact set .
Setting , we have . Thus we get . On the other hand, using (iii) Then we get , and hence it follows that , .
(b) We set so that and Then we get , . The conclusion follows taking the limit as . The proof of theorem is complete.
In the following theorem, we obtain a comparison result in terms of Lyapunov-like functions with ITD.
Theorem 14. Assume that(i), is locally Lipschitzian in , and (ii)the maximal solution of the fractional scalar differential equation exists for ;(iii) is nondecreasing in for each .Then implies
Proof. Define so that Let so that where . Since is locally Lipschitzian in and is the Lipschitz constant and as , we have By using Theorem 13 we obtain that
4.2. Stability and Boundedness Criteria
A comparison principle is obtained by employing the notion of Lyapunov function together with the theory of differential inequalities. In this part, one can see Lyapunov-like function as a transformation which reduces the study of stability, boundedness, and Lagrange stability properties relative to ITD of a given complicated system to a relatively simpler scalar equation.
4.2.1. Stability Criteria
Theorem 15. Assume that(i), is locally Lipschitzian in , and (ii)the maximal solution of (37) exists for ;(iii)there exists such that for ;(iv) is nondecreasing in for each and ;Then the stability properties of the null solution of (37) imply the corresponding initial time difference stability properties of the solution .
Proof. Assume that the null solution of (37) is equistable. Let be given. Then by definition of equistability given , , there exist a such that where is any solution of the (37). Choose as . Obviously . Then given and , and such that Let and choose . Then we claim that If it is not true, from (45), there exist a and a solution with and such that Moreover since , by (iii) we have Hence, by (i), (ii), (47) and, Theorem 14, we obtain the following estimate: Consequently, the relations (44), (47), (49), and (iii) lead to the contradiction which proves that the solution of (7) is equistable with ITD.
4.2.2. Boundedness Criteria
In this section, Lagrange stability, which includes boundedness criteria, of fractional dynamic systems are discussed by employing comparison method.
Proof. We need to prove (B1) and (S3) for (7). Let be given, and let . In view of (iii) . Assume that (37) is Lagrange stable. It follows that (37) is bounded. Then, given and there exists a such that
Moreover, as ; we can choose a verifying the relation
Then from (45), given and , and such that
Let . Then we claim that
If this were false, there would exist a and a solution of (7) such that . Consequently, the relations (49), (51), (52), and (iii) lead to the contradiction
which proves that (B1) holds for (7).
To prove attractivity, we let , , be given and . In view of (iii), . Since (37) is attractive in the large with ITD, given , and there exist a such that We have (49) from the proof of Theorem (45). Now, suppose that there exists a sequence , as , and a solution of (7) such that The relations (iii), (49), (56), and (57) lead to the contradiction, which proves that the system is attractive in the large with ITD.
Firstly, differential inequalities and a new comparison principle for fractional differential equations relative to initial time difference have been developed and then stability, boundedness criteria, and Lagrange stability relative to initial time difference have been proved by employing comparison method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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